Show that each of the relation R in the set `A={x in Z :0lt=xlt=12}`, given by(i) `R = {(a , b) : |a - b| ` is a multiple of `4}`(ii) `R = {(a , b) : a = b}`is an equivalence relation. Find the set of all elements related to 1 in each case.

Explanation

`A = {x € Z: 0 ≤ x ≤ 12}` = `{0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}`

(i) `R = {(a , b) : |a b| ` is a multiple of `4}`

For any element a ∈ A, we have (a, a) ∈ R as a - al = 0 is a multiple of 4

`:.` R is reflexive.

Now, let (a, b) ∈ R⇒ |a - b| is a multiple of 4.

⇒-(a - b) = |b - a| is a multiple of 4.

⇒ (b. a) ∈ R

`:.` R is symmetric.

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